MinT - Best Known (234−85, 234, s)-Nets in Base 4

Best Known (234−85, 234, s)-Nets in Base 4

(234−85, 234, 160)-Net over F₄ — Constructive and digital

Digital (149, 234, 160)-net over F₄, using

1 times m-reduction [i] based on digital (149, 235, 160)-net over F₄, using
- (u,Â u+v)-construction [i] based on
  1. digital (33, 76, 56)-net over F₄, using
    - net from sequence [i] based on digital (33, 55)-sequence over F₄, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 33 and N(F) ≥ 56, using
        F₅ from the tower of function fields by GarcÃa and Stichtenoth over F₄ [i]
  2. digital (73, 159, 104)-net over F₄, using
    - net from sequence [i] based on digital (73, 103)-sequence over F₄, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 73 and N(F) ≥ 104, using
        F₆ from the tower of function fields by GarcÃa and Stichtenoth over F₄ [i]

(234−85, 234, 196)-Net in Base 4 — Constructive

(149, 234, 196)-net in base 4, using

2 times m-reduction [i] based on (149, 236, 196)-net in base 4, using
- trace code for nets [i] based on (31, 118, 98)-net in base 16, using
  - 2 times m-reduction [i] based on (31, 120, 98)-net in base 16, using
    - base change [i] based on digital (7, 96, 98)-net over F₃₂, using
      - net from sequence [i] based on digital (7, 97)-sequence over F₃₂, using
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 7 and N(F) ≥ 98, using
        a function field by SÃ©mirat [i]

(234−85, 234, 466)-Net over F₄ — Digital

Digital (149, 234, 466)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(234−85, 234, 12007)-Net in Base 4 — Upper bound on s

There is no (149, 234, 12008)-net in base 4, because

1 times m-reduction [i] would yield (149, 233, 12008)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 191 186274 614098 759082 739944 157034 307765 204192 119350 134335 996205 437565 580692 935256 016826 397418 603765 759722 776098 837577 558661 687580 144906 920776 > 4²³³ [i]